Abstract
Assume that \({\mathfrak {H}}\) and \({\mathfrak {K}}\) are two real or complex Hilbert spaces, A a linear relation from \({\mathfrak {H}}\) to \({\mathfrak {K}}\), and B a linear relation from \({\mathfrak {K}}\) to \({\mathfrak {H}}\), respectively. Necessary and sufficient conditions for B to be equal to the adjoint of A are provided. Several consequences are also presented. More precisely, new characterizations for closed, skew–adjoint, selfadjoint, normal linear relations, and generalized orthogonal projections are obtained.
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Appendix
Appendix
The next result is useful with respect to the multiplication of \(2 \times 2\) matrices of linear relations.
Lemma 9.1
Assume that \({\mathfrak {H}}\) and \({\mathfrak {K}}\) are two real ar complex Hilbert spaces and let \(A \in {\mathfrak {L}}({\mathfrak {H}},{\mathfrak {K}})\) and \(A \in {\mathfrak {L}}({\mathfrak {K}},{\mathfrak {H}})\) The following identity holds true:
Proof
Assume that \( \left\{ \left( \begin{array}{c} x \\ y \end{array} \right) , \left( \begin{array}{c} z \\ t \end{array} \right) \right\} \in \left( \begin{array}{cc} I &{}\quad -B \\ A &{}\quad I \\ \end{array} \right) \left( \begin{array}{cc} I &{}\quad B \\ -A &{}\quad I \\ \end{array} \right) \), so that
for some \(a \in {\mathfrak {H}}\) and \(b \in {\mathfrak {K}}\). Therefore, \(a = x + y^{\prime }\) and \(b = -x^{\prime } +y\) for some \(\{x,x^{\prime }\} \in A\) and \(\{y,y^{\prime }\} \in B\). Also \(z = a-b^{\prime }\) and \(t = a^{\prime } + b\) for some \(\{a,a^{\prime }\} \in A\) and \(\{b,b^{\prime }\} \in B\) . Consequently, \(z = x + y^{\prime } - b^{\prime }\) and \(t = a^{\prime } - x^{\prime } + y\). Furthermore, \(\{x,x^{\prime }\} \in A\) implies that \(\{x,y-b\} \in A\) and:
so that \(\{x,y^{\prime }-b^{\prime }\} \in AB\), which further leads to:
Similar arguments shows that:
Since also:
it follows from (9.13), (9.14) and (9.15) that:
Thus:
Conversely, it will be shown that:
Assume that \( \left\{ \left( \begin{array}{c} x \\ y \end{array} \right) , \left( \begin{array}{c} z \\ t \end{array} \right) \right\} \in \left( \begin{array}{cc} I + BA &{} \mathrm{dom\,}B \times \mathrm{mul\,}B \\ \mathrm{dom\,}A \times \mathrm{mul\,}A &{} I + AB \\ \end{array} \right) , \) so that \(z = x^{\prime } + m_{2}\) and \(t = y^{\prime } + m_{1}\) for some \(\{x,x^{\prime }\} \in I + BA\), \(\{y,y^{\prime }\} \in I + AB\), \(m_{1} \in \mathrm{mul\,}A\) and \(m_{2} \in \mathrm{mul\,}B\). It follows from \(\{x,x^{\prime }\} \in I + BA\) that \(\{x,x^{\prime }-x\} \in BA\), so that \(\{x,a\} \in A\), \(\{a,x^{\prime }-x\} \in B\) for some \(a \in {\mathfrak {K}}\). Also, it follows from \(\{y,y^{\prime }\} \in I+AB\) that \(\{y,y^{\prime }-y\} \in AB\), so that \(\{y,b\} \in B\), \(\{b,y^{\prime }-y\} \in A\) for some \(b \in {\mathfrak {H}}\). Then, clearly:
Furthermore, the following relations:
imply that:
A combination of (9.18) and (9.21) shows that:
so that (9.17) has been proved. Finally, it follows from (9.16) and (9.17) that (9.12) holds true. \(\square \)
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Sandovici, A. On the Adjoint of Linear Relations in Hilbert Spaces. Mediterr. J. Math. 17, 68 (2020). https://doi.org/10.1007/s00009-020-1503-y
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DOI: https://doi.org/10.1007/s00009-020-1503-y
Keywords
- Hilbert space
- closed linear relation
- Skew–adjoint linear relation
- selfadjoint linear relation
- normal linear relation
- generalized orthogonal projection